We describe the action of the fundamental group of a closed Finsler surface of negative curvature on the geodesics in the universal covering in terms of a flat symplectic connection and consider the first order deformation theory about a hyperbolic metric. A construction of O.Biquard yields a family of metrics which give nontrivial deformations of the holonomy, extending the representation of the fundamental group from SL(2,R) into the group of Hamiltonian diffeomorphisms of S^1 x R, and producing an infinite-dimensional version of Teichmuller space which contains the classical one.
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